I'm still wondering why space is 3-dimensional.
The mathematics that model space fit conveniently in 3 dimensions without becoming unmanageably complex.
However, by employing additional dimensions, particularly dimensions in which the basis vector multiplied by itself is a product other than itself, you can sometimes simplify the math that describes portions of the universe to a shocking extent. For instance, using geometric algebra with one dimension that squares to positive one and three that square to negative one, Maxwell's Equations reduce to "nabla field_bivector = free_space_permeability * speed_of_light * current_vector".
Sometimes, introducing special-purpose dimensions with interesting geometries and constraints upon the multidimensional representation makes certain types of math easier. For instance, conformal representation employs two dimensions that square to zero to represent the origin and infinity, and regular space is represented as a curved subset of the hyperdimensional space. Otherwise complicated operations become simpler, as a translation and rotation through dimensions that do not exist that almost coincidentally lands the result right back on the constrained hypersurface that represents 3-dimensional space.
When we invent these extra dimensions for the purposes of doing the math, we have no good way of knowing whether they are entirely imaginary or just undetectable and inaccessible to us at our current level of technology. If the universe had a round dimension with a diameter of less than the Planck distance, we would not be able to detect it, because we can't measure a distance that small. But maybe certain properties in the particle zoo can be more easily explained using an angular phase value on a tiny round dimension perpendicular to everything else.